We represent knot sequences as random walks
on a triangular lattice. The axes *r*,*c*,*l* correspond to the three
regions *R*,*C*,*L* through which the active end can be wound and the unit
vectors represent the corresponding
moves (Fig. 3); we omit the directional notation and
the terminal action *T*.
Since all knot sequences end with and
alternate between and , all knots of odd half-winding number
*h* begin with
while those of even *h* begin with .
Our simplified random walk notation is thus unique, and we only make use of
the directional notation in the context of move sequences.

The three-fold symmetry of the move regions implies that
only steps along the positive lattice axes are acceptable and, as in the case of
moves, no consecutive steps can be identical; the latter condition makes
our walk a second order Markov, or persistent, random walk.
Nonetheless, every site on the lattice can be reached
since, *e.g.*, and
.

**Figure 3:** A tie knot may be represented by a persistent random walk on a
triangular lattice, beginning with and ending with
or . Only steps along the
positive *r*,*c* and *l* axes are permitted and no two consecutive
steps may be the same.
Shown here is the Four-in-Hand, indicated by the walk
.

The evolution equations for the persistent random walk appear as

where is the conditional probability that the walker is
at point (*r*,*c*,*l*) on the *n*th step, having just taken a step along the
positive *r*-axis, *p* is conditioned on a step along the positive *c*-axis,
*etc*. The unconditional probability of occupation of a site, *U*,
may be expressed .

Sat Nov 7 16:03:57 GMT 1998